On global well-posedness of the modified KdV equation in modulation spaces

Sharp Global Well - Posedness for Kdv and Modified Kdv On

The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all L 2-based Sobolev spaces H s where local well-posedness is presently known, apart from the H 1 4 (R) endpoint for mKdV. The result for KdV relies on a new method for co...

Sharp Global Well-posedness for Kdv and Modified Kdv on R and T

LOCAL WELL-POSEDNESS FOR THE MODIFIED KDV EQUATION IN ALMOST CRITICAL Ĥr

We study the Cauchy problem for the modified KdV equation ut + uxxx + (u )x = 0, u(0) = u0 for data u0 in the space Ĥr s defined by the norm ‖u0‖Ĥr s := ‖〈ξ〉 sû0‖Lr′ ξ . Local well-posedness of this problem is established in the parameter range 2 ≥ r > 1, s ≥ 1 2 − 1 2r , so the case (s, r) = (0, 1), which is critical in view of scaling considerations, is almost reached. To show this result, we...

On the Well Posedness of the Modified Korteweg-de Vries Equation in Weighted Sobolev Spaces

We study local and global well posedness of the k-generalized Korteweg-de Vries equation in weighted Sobolev spaces Hs(R) ∩ L2(|x|2rdx).

Remark on Well-posedness and Ill-posedness for the Kdv Equation

We consider the Cauchy problem for the KdV equation with low regularity initial data given in the space Hs,a(R), which is defined by the norm ‖φ‖Hs,a = ‖〈ξ〉s−a|ξ|a b φ‖L2 ξ . We obtain the local well-posedness in Hs,a with s ≥ max{−3/4,−a − 3/2}, −3/2 < a ≤ 0 and (s, a) 6= (−3/4,−3/4). The proof is based on Kishimoto’s work [12] which proved the sharp well-posedness in the Sobolev space H−3/4(R...